Hyperloop: riding sound’s density peak to exploit the drag equation

Super short version: if you ride along with a low enough frequency soundwave at the speed of sound, you can trade reduced incoming particle velocity for increased incoming particle density; the drag equation favors this heavily as it has both a density factor and a velocity squared factor; take this trade to an extreme with a high enough amplitude sound wave and you get nearly-adiabatic travel between point A and point B at the speed of sound

Hyperloop is Elon Musk’s proposed means of travel that could enable a trip between San Francisco and LA in 30 minutes (which would mean you travel at roughly the speed of sound). He is supposed to be revealing an “alpha” design in August.  I think I have finally worked it out what it will be, and passengers should find themselves travelling at exactly the speed of sound.

The method I propose would enable extremely low drag (i.e. air resistance) at sea-level altitude without doing anything that has to affect the air across the whole loop (e.g. evacuating the entire tube or circulating the entire column of air). The key is a combination of a scaling property of the high-velocity drag equation and a clever way to exploit it by traveling within the density peak of an intense low-frequency sound wave. 

Hyperloop is mainly just a looping tube (with entry/exit accelerators/decelerators) that acts as a waveguide for these bursts of soundwaves:


The properties of sound waves get interesting when you view them from the reference frame of a vehicle embedded within, traveling in the direction of the wave at the speed of sound.  The vehicle remains fixed on one point of the pressure wave as the wave propagates.  Depending on the point along the wave you chose, air density and air velocity differ inversely, allowing  net airflow to always remain the same (as required if you are to move through the air; sound waves themselves just jiggle air back and forth and don’t actually move it anywhere in the long run).


Traveling in the wave’s trough, a vehicle would face a sparse number of molecules whizzing by at high velocities, but riding the peak, the vehicle would only have to deal with a dense cloud of molecules slowly crawling by.  The latter turns out to involve much less drag over the course of a hyperloop trip!

Fd = (½)P*(V^2) CdA

the high velocity drag equation, Fd = (½)P*(V^2) CdA  

The force of drag increases proportionately as the square of the air velocity (v^2), but only linearly with air density (ρ)!  If you double the density, you only double the drag, but if you instead double the velocity, you quadruple the drag.  The drag equation is ultimately this way because incoming particles impart a momentum of only m*v on the craft but a kinetic energy of m*v^2.  In the dense part of the wave there is a high potential energy, but low kinetic energy (with respect to the net velocity of the air).  In the sparse part the potential energy has been converted into net velocity (temperature goes down and velocity goes up as the gas expands).

This in turn means that when traveling along with the sound wave, all the vehicle needs to do to minimize drag is stay on the wave peak–the area of highest density and lowest apparent velocity.  And the higher the amplitude of the wave hyperloop manages to generate, the higher the density, the slower the apparent air velocity, and the lower the drag during the trip!  Crank it up enough and you approach fully adiabatic travel.

So, in the launching area the vehicle reaches the speed of sound and is then ejected into the main loop at exactly the right time to be at the peak of a passing sound wave.  All it then needs to do is use a low amount of force to fight the drag induced by the dense soup of slow, incoming air  (slow from the vehicle’s reference frame) for around 30 minutes or so, and BAM! it has traveled from San Francisco to Los Angeles.  That’s it; that’s hyperloop.  You shouldn’t really need to read the rest.

But, I’ll quickly go over how this version of Hyperloop can meet the conditions hinted at by Elon Musk:

  • It lets you travel from LA to San Francisco in around 30 minutes

  • It is like a cross between a Concorde, a railgun, and an air-hockey table

  • It isn’t affected by earthquakes

  • It could use solar to be self-powering

  • It can store excess solar energy within the system itself

  • You can’t crash


It lets you travel from LA to San Francisco in around 30 minutes:

This is roughly the speed of sound; everything I’ve mentioned depends on moving at this speed.


It is like a cross between a Concorde,

Sonic-ish speeds and lots of noise?


…a railgun,

I think acceleration/deceleration will be done in a separate area from the main loop; if the vehicles were accelerated in the main tubes there would be the possibility of wave reflectance if a wave passed by before it was up to speed.  Anyway, this could be done with a railgun or a coilgun; I suspect coilgun because that would also be an easy design to use for maintaining vehicle velocity with solar during the main stretches of the loop.


…and an air-hockey table

You could scoop in some of the dense air and blow it through air hockey holes at the bottom of the craft…  If the pressure is high enough at the soundwave peak, you could design the craft to be neutrally buoyant (I think we’d be getting into shock wave territory by that point though).


It isn’t affected by earthquakes

I believe some fragile historic buildings have been anchored to a slab that rests atop a second slab anchored to the ground; in an earthquake the slabs just slide freely over each other and the structure doesn’t feel much force.


Same deal here; anchor each Hyperloop tube-support to a sliding slab.


It could use solar to be self-powering

PV solar would be used to power the rail/coilguns that maintain the vehicle’s speed in the low drag environment.  Solar would also be used to create and maintain the sound wave; maybe it could use cheap thermal solar along with PV to create waves in the manner of a thermoacoustic engine.


It can store excess solar energy within the system itself

The waves travelling throughout the system would be the energy storage.  Friction losses between air and the tube walls could be mitigated by just increasing the diameter of the tube: the perimeter grows proportional to the diameter, but the encircled air grows as the diameter squared.  So that scales well.  Lowering the wave frequency in the design could minimize losses as well by making the waves more and more adiabatic in nature.


You can’t crash

Famous last words…


35 responses to “Hyperloop: riding sound’s density peak to exploit the drag equation”

  1. Hayden says :

    exactly how much sound are we talking about to propel a cabin full of people?

    • charlesalexander2013 says :

      The sound doesn’t actually do the propelling. You reach the speed of sound through other means, and then ride along with the sound wave: at some points (from a speed of sound reference frame) it slows down the apparent wind at the cost of increased density, or at other points along the curve it speeds up the apparent wind in exchange for less density.

      You still have to use energy to fight this apparent wind and maintain speed of sound, the post tries to show that it is easier to do so in the increased density case.

  2. wtpayne says :

    That was an awesome post:- There is some really cool physics here!

    I still do not understand how the craft maintains it’s forwards speed, though.

    Presumably, it would need an impeller or something similar to shovel the high density air from the front to the rear? (and bottom) I assume we are talking about *really* high density air – to the point where it behaves a bit more like water than the air that we are used to?

    Or perhaps something special happens when the craft almost fills the tube? The craft body would also transmit the pressure wave – could it make the sound wave act like the peak density -> infinity? (speculation). Does this make a difference?

    • charlesalexander2013 says :

      I think I only had one mention kinda obscurely into the post, but my best guess is the entire main loop is an induction coilgun: https://en.wikipedia.org/wiki/Coilgun

      This would match up well with Elon’s mention of the loop having solar throughout.

    • strangemonad says :

      “I still do not understand how the craft maintains it’s forwards speed, though.”

      An object at rest stays at rest, an object in motion stays in motion unless acted on by a force. You’d need a couple boost along the way from magnetic coils or something so you don’t gradually fall out of the wave but other than that, while you’re in the wave you have little resistance

  3. Donhodges says :

    I like your last sentence – what could possibly go wrong? If Musk succeeds with Tesla and SpaceX he might just pull it off.

  4. Paul Reinheimer says :

    Wasn’t one of the other promises that there wouldn’t be any issues with right-of-way (or the massive amounts of land required)?

    Unless they’re building it down the freeway, this design seems problematic.

    • elroysf says :

      It’s being offered as an alternative to the multi-billion dollar CA high-speed rail project which cuts through hundreds of miles of prime central valley farmland is *highly unpopular* with farmers to say the very least. This system, would at least be able to be elevated above the land and leave farms largely intact.

    • thrackle says :

      He has said in interviews it could run down the center of highways, thereby avoiding right of way issues.

  5. ibrahim says :

    So this will be a quiet vehicle that travel at the speed of sound. This all makes sense in an equation on paper, but I can’t wait to see how it will operate in the real world where there is always an unpredictable factor.

  6. Mike says :


    Athough simple radio waves are interesting my advice would be to look at light. DWDM is the mechanics that multiplex light wave freq on a single mdium mich smaller than the entire overall air mass.


  7. Seager Mason says :

    Very interesting! Does this mean that each module is in effect just surfing the sound wave?

    • charlesalexander2013 says :

      Not quite; the wave never pushes the craft and the craft experiences drag the entire time. What this does is let the craft trade away incoming air velocity in exchange for higher air density; a thick, slow apparent wind turns out to cause much drag than a thin, fast apparent wind.

  8. ESRogs says :

    Could you elaborate on the velocity and pressure varying inversely part? I see from the first equation here: http://en.wikipedia.org/wiki/Sound_intensity#Acoustic_intensity, that for a given instantaneous intensity, pressure and velocity would have to vary inversely, but I didn’t see anything indicating that the intensity would be constant across a whole wavelength of the wave.

    Is there another set of equations I can look at to understand the relationship?

    • charlesalexander2013 says :

      I basically just did a flow analysis and didn’t bother with many equations (mainly just because I never had enough calculus classes to get to where they covered partial differential equations). I just knew since sound doesn’t permanently displace anything, there were a lot of restrictions on flow.

      Say you had a 100 meter tube loop with a million air particles. If you looped around the tube once at any speed, you would pass by a million particles (ignoring thermal dispersion, because it averages out to zero). And the same thing would hold in the presence of a sound wave (as long as it had a wavelength much smaller than the tube length), because sound doesn’t permanently displace any particles.

      I don’t think that intensity equation you linked holds constant in the normal reference frame; the sign flips as the velocity oscillates if nothing else. And there is a mathematical singularity at the point where avg particle velocity reaches 0 =).

      • esrogs says :

        Oh right, so the idea is that whatever part of the wave you’re riding along in, you have to pass the same number of particles per unit time. So if they’re passing you relatively quickly they must be sparser, and if relatively slowly they must be denser. I think part of my confusion was mixing up pressure and density (though, would the most dense part of the wave also be the pressure peak?).

        An additional issue is — how does the presence of the pod affect the sound wave? You now have to have air molecules flowing around the sides of the pod, and turbulence behind it (according to wikipedia, a requirement of the applicability of the drag equation). Any idea whether this disrupts the propagation of the wave?

  9. Mohammed AlQuraishi says :

    What is the maximum achievable air density? That should put some bounds on how much energy would be required to stay at the speed of sound for a 30-min trip.

  10. Bryan Baker says :

    So the air in the tube is 1atm and at rest. Produce a very low frequency, high amplitude sound. Low frequency means the size of the RAREFIED/compressed zones match the size of the capsule. The sound wave & capsule will both be traveling at the speed of sound with respect to the ground/medium/still air, and traveling at 0mph with respect to each other. The capsule will travel in the rarefied section.

    I think you may be confusing speeds relative to the ground with speeds relative to each other?

    • charlesalexander2013 says :

      No, the capsule won’t travel in the rareified section; that was my initial thought but that ends up even worse than traveling through undisturbed air because of the incoming velocity of the particles in that section (see the particle flow analysis and the shitty diagram of the capsule). Instead it rides along the most dense part of the wave and then through the drag equation you get a discount because of the low apparent velocity of incoming particles. Velocity*mass (momentum) of the incoming particles is always the same along any point of the wave (from the vehicle’s reverence frame), but velocity*mass^2 (kinetic energy) can change.

      I still haven’t done all the numbers.. don’t think it will turn out practical; like the temperature in the dense region may be so hot that you can’t keep the vehicle cooled enough over the total travel time. Or the wavelength might have to be so large to avoid non-linear scaling at high pressures that it takes so much energy to setup and maintain that it just doesn’t buy you enough, or won’t even fit in the entire loop or something.

      • Bryan Baker says :

        But if the dense region has low velocity with respect to the ground, then it has a velocity of 760mph with respect to the capsule traveling at 760mph with respect to the ground. You want to place the capsule in that section of the wave form where the average air particle velocity is at its maximum with respect to the ground, not zero (or worse, -760mph) with respect to the ground.

      • esrogs says :

        “but velocity*mass^2 (kinetic energy) can change”

        You meant, velocity^2*mass, right?

      • charlesalexander2013 says :

        Bryan: Read the flow analysis above that shows that there must be the same flow at all points when you are going the same speed as the wave propagation speed. If at any point there was a differing net flow at any point, particles would have to be going somewhere.

        This isn’t quite true from a resting reference frame, because particles may pass you back and forth. But if you move from one side of the tube to the other, you will pass all the particles regardless of your speed (ignoring thermal dispersion).

        With the flow analysis showing that, from the reference point of the vehicle travelling with the wave, flow must be the same at all displacements, we know that displacements from the vehicle where the air is denser must have a lower average incoming velocity from the places where it is sparser.

        In fact if you see a static snapshot of a wave, having the velocity vectors available to look at in those regions is the only way to determine the directionality of the wave.

        Think of a speaker creating a wave, it pushes air as the wave climbs to a crest and makes it denser, then it pulls it back and makes it sparser, repeatedly.

        Or stare at this picture for a for a while: http://www.acs.psu.edu/drussell/demos/waves/wavemotion.html

        You might be getting confused about wave propagation speed vs. the actual peak velocities imparted on the particles. Those velocities usually don’t get anywhere near the speed of sound and when they do you have a sonic boom/shockwave, not a regular adiabatic sound wave.

        esrogs: yes. velocity^2 * mass.. I’m not even too sure about the reasoning there so I’m just going to stick with the drag equation to be safe; but I think the argument goes something like this: you are building up potential energy when things get dense and hot, so the particles must lose net velocity and therefore lose kinetic energy, and then you are gaining net velocity as the air expands back out, by converting the potential energy of the pressure / temperature in the compressed air. I think this also explains why sound propagation is roughly adiabatic.

        When I say “potential energy” I know that heat is stored as kinetic energy in the particles, but since their vector directions contributed by heat are random and net out to zero, from a macroscopic perspective “potential energy” is as good a word as any.

      • Bryan Baker says :

        You’ve drawn about 1.33 of a complete waveform. So how can ALL of your air particles velocity vectors (with respect to the ground) be negative in direction? Each particle moves +x from its original position, then -x from original. So there should be particles doing +v (with respect to the ground), -v (with respect to the ground), and everything in between.

        If on this graph, http://physics.info/waves/longitudinal-wave.html, you were to plot the average particle velocity (with respect to the ground) alongside pressure on this graph, would it not be the inverse of the pressure curve?

        Bernoulli’s says high speed = low pressure.

      • esrogs says :

        Bryan, the particles have negatively velocity vectors with respect to the pod, not with respect to ground. WRT ground the velocities have mixed sign, just as you describe.

      • charlesalexander2013 says :

        There is a caption that says the reference frame is moving towards the right at the speed of sound. So the velocity vectors are with respect to that reference frame. The arrows kind of look like they get to zero in that reference frame at some point, but it shouldn’t and they don’t.

  11. thrackle says :

    Awesome post, Charles.

    I’m setting up a community site for people interested in the hyperloop. Hope you don’t mind if I mention it here. It is:


    I just think this is a really neat idea, and I wanted to get involved and help some way.

    I’m completely open to stepping back and letting the community run the site, so if anyone here wants to get involved, just let me know.

  12. Cliff Fornwalt says :

    What about reflections of the sound waves and flow of gas around the pod itself? If the pod is large relative to the internal diameter of the tube (i.e., a small gap), you would expect the individual gas particles (with negative or positive vectors relative to the pod) to have to accelerate and pass through this constriction point. You said yourself that the pod would still have to pas every gas particle in the tube in your model. Your model only works well if the pod is very small in a very large tube. I don’t think Elon is planning pod diameters of 2m and tubes 10m across – but maybe.

    Doesn’t it make more sense to use a low viscosity gas and blow it around the tube at something close to your desired travel speed?

    Just fill the tube with helium. This would lower the viscosity significantly.

    Then, use millions of tiny holes (like an air-hockey table) to draw off the boundary layer (see this wikipedia page about boundary layer suction: http://en.wikipedia.org/wiki/Boundary_layer_suction) to improve laminar flow as gas moves around the tube. This would even further reduce drag in the system.

    A double-walled tube (think of a suppressor for a gun) with backflowing air/helium in the intertube space would create a natural venturi effect to suck to boundary layer off. You could even blow some of this gas out holes under the pod to create an air-hockey-like levitation mechansm.

    I imagine you could combine all these features with your concept of a standing wave in a long looped tube – I don’t see them as mutually exclusive – but all of them together might acheive the efficiency needed for this project.

    • charlesalexander2013 says :

      Yes, vehicle diameter would have to be some fraction of tube diameter for there to be room to flow.. otherwise the craft is just closer and closer to a piston. I still drew it too large, but in the picture the diameter is still about 2x the pod.

      Anyway, this idea is pretty much shot after reading elsewhere that hyperloop does supersonic travel. It might be due to helium, but that would take a lot of party balloons. I would think you would just evacuate the tube at that point.

      I never ran any numbers, just made sure everything worked as long as you variously scaled different things, even if it would end up being comically large and if the temperature in the high-pressure region is so hot it would bake the passengers.


      I’m thinking now about ways it could work by redirecting shock-waves produced on the front-end of the pod and reabsorbing them on the tail-end. The tail would be shaped like a V, with shock waves coming in from the sides like this:

      –> V <–

      Which pushes the pod up as the waves get reflected downward.

      It wouldn't work with a normal tube (you could only reflect the lateral momentum, not the vertical momentum), you'd have to have wave-guides enveloping the outside of the tube to redirect the shock waves from travelling forwards to traveling inwards.

      Some supersonic planes do something similar to redirect the shock waves to the rudder and make it easier to steer at mach speeds, but they can't gain any forward momentum from any tricks like that without having fixed structure to reflect off of.

      I have no guesses on how excess energy could be stored in the day and used overnight in this scheme…

  13. Cliff Fornwalt says :

    Oh my… That’s the answer… The big problem is how can you travel 600 mph “down the center of freeways?” The turns would be impossible for humans to bear. The answer is that the tube is very large compared to the size of the pod… The pod floats on a cushion of air (air hockey table), and naturally banks up the wall in turns. All the G forces remain downward-pointing. It also makes it more compatible with your standing wave idea.

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